Strain Energy and Modulus Of Resilience

Definition of strain Energy and Modulus Of Resilience.
Development Of Strain Energy Formulae.
Computation of the Strain Energy and Modulus of Resilience Of Engineering Materials.

1. STRAIN ENERGY By L.E EZE

2. S trength of Material Module 1 : Strain Energy Lecture: 03 Topic : Strain Energy and Modulus Of Resilience Learning Outcomes Definition of strain Energy and Modulus Of Resilience Development Of Strain Energy Formulae Determine the Strain Energy and Modulus of Resilience Of Engineering Materials

3. STRAIN ENERGY(U) Strain energy is the internal energy which is stored in any material that is loaded within its elastic limit. It is also the work done by a loaded material . it is also called RESILIENCE. Strain Energy (U) =   Resilience is the energy stored within Elastic limit. It is given by ‘U’ Mathematically ,

4. Development of strain energy formula Consider the bar rod below; Axially loaded member Stress stored in the bar δ =   Strain( ε ) =   Young modulus(E) =   Young modulus(E) =   Re-arranging equation 1 ; (E) = × L   1 e = ×L  

5. But, Hence; Stress stored in the bar δ =   If we plot a graph of load against extension for the axially loaded rod, we shall obtain a straight line graph. e =   2

6. Area of the graph = Energy stored Area of shaded portion =   Area = A .   Area = A . . .   But, δ =   Recall: P = A and e =   Area of the shaded portion = Area of a triangle Hence ; A rea Load extension

7. Area of the graph = Energy stored Alternatively, Substitute e = To obtain;   Therefore ; Area Of graph = .   A L = Volume   Area Of graph = .V   Energy stored = Strain Energy(U) Area Of graph = .   Strain Energy(U) = .V   3 Strain Energy(U) = .   4

8. Proof Resilience (U max ) = .V   mathematically ; Proof Resilience : it is the maximum strain energy stored in a material. Proof resilience simply means (maximum strain-energy). Modulus of Resilience =   Modulus of resilience is calculated as area under the stress- strain graph Modulus of Resilience: it is defined as the strain energy per unit volume. Prove Of M.O.D : M.O.D =   5

9. Area of the graph = × ε   Area = × ×   Area of the graph = M.O.R ×P.e = Area under the load-extension graph = Strain Energy   ε = δ =   But: stress A rea strain A× L = Volume M.O.R = ×   6

10. Hence; Area of bar = 500 mm 2 = 0.0005m Length of bar = 600mm = 0.6m P = 50kN =50,000N E = 200GN = 200×10 9 N/ m 2 Solution Invoking equation 2 Example 1: A prismatic bar of cross –section 500mm 2 length 600mm is acted upon by a 50kN force. Determine; The total elongation of the bar The strain Energy Consider E= 200GN/ m 2 M.O.R =   7 600mm 50kN 50kN e =   2

11. e =   U = U = 7.5J   e =   e = 0.0003m Strain Energy(U) =   Applying equation 4 ANSWERS Total elongation bar = 0.0003m Strain Energy in the bar = 7.5J

12. Area of bar = 0.25m × 0.45m Length of bar = 3.5m P = 2000N E = 3MN/m 2 = 3×10 6 N/ m 2 Solution Example 2: A prismatic bar 3.5m long is subjected to axial compressive force 2000N. If the cross- sectional area of the bar is 0.25m × 0.45m, and modulus of Elasticity is 3MN/ m 2 calculate; The strain Energy in the bar The modulus of Resilience 3.5m 2000N 2000N Invoking equation 4 4 Strain Energy(U) =  

13. U = U = 20.741J   ANSWERS Applying equation 7 Strain Energy in bar = 20.74J Modulus Of Resilience = 52.67N/ m 2 M.O.R =   Strain energy = 20.741J Volume = area × length of bar Volume = 3.5 × 0.25 × 0.45 Volume = 0.39375m 2 M.O.R =   M.O.R = 52.67N/ m 2

14. Area of bar = 4 m 2 Length of bar = 10m P = 4000N E = 3000N/m 2 Solution Invoking equation 4 Example 3: A prismatic bar 10m long is subjected to axial compressive forces P = 4 000N. Compute the amount of U stored in the bar if the cross-sectional is 4m 2 , and M.O.E is 3,000N/m 2 10m 4 000N 4 000N 4 Strain Energy(U) =  

15. U = U = 6666.67J   ANSWERS Strain Energy in bar = 20.74J

16. Reference Books 1 . Robert L .Mott & Joseph A Untener –Applied Strength Of Material 2. R.k. Bansal – Strength Of Material 3. R.k Rajput – Strength of material

17. L.E Eze Teacher Scientist Sismo Academy [email protected] Thank you